Sometimes when we want to work out the area under a curve, we cannot integrate the function easily.  In these cases, we can use numerical integration methods to calculate the area.  Numerical methods are not exact methods; rather, they approximate the area as being, for example, a whole lot of rectangles.

Riemann Sums

One way of working out a fairly good approximate value for the area underneath a curve is to represent the area as lots of thin rectangles.  This technique is known as a Riemann Sum.

The general form of a Riemann Sum is:

Say I want to calculate the area beneath the function  from x = 0 to x = 5.

Firstly, I must decide how many rectangles I want to use to simulate the area.  The more rectangles I use, the more accurate the answer, but the longer it will take.  Let’s use 5.

Since the interval is from 0 to 5, and we are using 5 rectangles, each rectangle must be 1 unit wide – Dx = 1.

The mathematical sign with the ‘i=1’ below it is called a sigma sign.  All it means in this case is:

·         Calculate the value of the function at x = 0 and multiply it by Dx.

·         Calculate the value of the function at x = 0 + Dx and multiply it by Dx

·         Calculate the value of the function at x = 0 + 2Dx and multiply it by Dx.

·         Keep doing this until you get to x = 5 – Dx.

·         Add all these results together.

So:

For :

For :

For :

For :

For :

Adding these results together, you get 115 as the area under the curve between x = 0 and x = 5.